History of Mathematics

The principle of virtual work

In every equilibrium of any forces whatever, in whatever way they are applied, the sum of the positive energies will be equal to the sum of the negative energies. — Johann Bernoulli, letter to Pierre Varignon (26 January 1717) Since Archimedes, each simple machine had had its own explanation. The lever obeyed the law of the arms; the inclined plane, a decomposition of forces invented by Stevin; the pulley, an argument about the ropes; the screw, yet another. In 1717, in a letter to his French friend Pierre Varignon, Johann Bernoulli showed that all these laws are cases of a single principle.

October 13, 2026 Read

Chladni figures and the prize for vibrating plates

Sprinkle sand on a metal plate held at its centre and draw a violin bow across the edge. The plate begins to sing, and the sand, as if it had a will of its own, flees some places and gathers in others, drawing lines, crosses, stars and symmetrical curves. Change the note, and the drawing changes. These are Chladni figures. How a Chladni plate is excited: the plate is held at the centre and a violin bow is drawn across the edge. Engraving from an 1879 book.Unknown artist, 1879 · Wikimedia Commons · public domain Sound made visible The German Ernst Chladni published these figures in 1787, in a book called Discoveries on the Theory of Sound. The qualitative explanation is simple. When the plate vibrates, almost all its points move up and down and throw the sand away. But some lines stay still: these are the nodal lines. The sand slides there and stays. The drawing is a map of the places where the plate does not move.

October 13, 2026 Read

Euclid's algorithm: the greatest common divisor

The Euclidean algorithm is the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day. — Donald Knuth, The Art of Computer Programming, vol. 2 (1969) You have two rods, one 252 cm long and the other 105 cm, and you want the longest ruler that measures both exactly, a whole number of times. In today’s language you want the greatest common divisor, $\gcd(252,105)$. You could list all the divisors of each number and look for the largest common one, but with big numbers that is impractical. In Book VII of the Elements, Euclid teaches a much faster way.

October 12, 2026 Read

General relativity: from the lift to the Sobral eclipse

I was sitting in a chair in the patent office in Bern when all of a sudden a thought occurred to me: if a person falls freely, he will not feel his own weight. I was startled. This simple thought made a deep impression on me. It impelled me towards a theory of gravitation. — Albert Einstein, on an idea of 1907 The relativity of 1905 had a flaw: it did not include gravity. Newton’s law says the Sun pulls the Earth instantly, across 150 million kilometres, and relativity says nothing travels faster than light. Einstein spent eight years trying to fix this. His starting point was what he called “the happiest thought of my life”.

October 12, 2026 Read

The golden theorem: the law of large numbers

Even the most foolish of men, by some instinct of nature, by himself and without any instruction, is convinced that the more observations have been made, the less danger there is of missing the mark. — Jakob Bernoulli, Ars conjectandi (1713), part IV With a die, we know the chance of each face in advance: $\frac16$. But what is the chance that a man of 40 will live another twenty years? That it will rain tomorrow? That a medicine will work? In these cases there are no equal faces to count. Jakob proposed another route: observe many times and use the frequency as an estimate of the probability. If, of many men aged 40, half reached 60, the chance must be close to one half.

October 11, 2026 Read

The mystic hexagram: six points on a conic

In February 1640 a sheet printed on one side only, like a poster, circulated in Paris under the title Essay on conics. It had no proofs, only definitions, three lemmas and a list of results. The author was 16 years old and his name was Blaise Pascal. When Father Mersenne showed the sheet to Descartes, the reaction was suspicion: Descartes thought the young man had taken everything from Girard Desargues, the architect and geometer Pascal admired. Desargues’s influence was real, and Pascal himself acknowledged it. But the central result on the sheet was new.

October 8, 2026 Read

Fermat and Pascal: fairly dividing an interrupted game

Two players bet the same amount on a game of several rounds: whoever first wins a certain number of rounds takes everything. Midway through, the game has to be stopped. How should the money be divided fairly? This is the problem of points. It looks simple, but it defeated mathematicians for more than 150 years. The wrong answer In 1494 the Italian friar Luca Pacioli proposed dividing the prize in proportion to the rounds already won. If A has won 2 rounds and B has won 1, A takes $\frac23$ of the prize.

October 7, 2026 Read

Leibniz's machine: turning sums into motion

In 1642 the young Blaise Pascal built a machine to help his father, a tax collector. The Pascaline added and subtracted by turning gears. To multiply, however, you had to repeat the addition by hand, many times. Leibniz saw a Pascaline in Paris and wanted to go further: a machine that would multiply and divide on its own. Multiplying is adding and shifting The trick is the same as in long multiplication at school. To multiply $23\times14$, we split 14 into $4+10$:

October 6, 2026 Read

The laws of motion: forces and changes in velocity

For Aristotle, a body moves only while something pushes it. Stop pushing the cart and it stops. It seems obvious, and it was accepted for almost two thousand years. Galileo had his doubts. Rolling balls down inclined planes, he noticed that the smoother the surface, the further the ball went. With no friction at all, it would never stop. Descartes reached a similar idea. Newton put these clues together and, in the Principia of 1687, wrote three laws that became the foundation of all mechanics.

October 6, 2026 Read

Counting prime numbers

Euler proved that the primes never end (see his chapter). But how many are there? At 15 years old, Gauss He spent his free time counting cousins in tables, one thousand at a time. Did you notice that, close to a number $x$, more or less 1 in every $\ln x$ number is prime. Hence the estimate: the number of primes up to $x$, which we call $\pi(x)$, is similar to

October 1, 2026 Read

The kissing circles theorem

In 1643’s letter to Princess Elisabeth, Descartes dealt with four tangent circles two by two. Call the curvature of a circle the inverse of the radius, $k = \frac{1}{r}$. The theorem says that the curvatures satisfy $$(k_1 + k_2 + k_3 + k_4)^2 = 2\left(k_1^2 + k_2^2 + k_3^2 + k_4^2\right).$$ Application: three equal coins of radius 1 touch each other two by two. What is the radius of the circle that fits in the hole between them? With $k_1 = k_2 = k_3 = 1$ and $k_4 = k$:

September 30, 2026 Read

The spring and differential equations

A differential equation is an equation in which the unknown is a function, and the equation speaks of its derivatives. Euler organized this path into a scheme that we still use today: observe a phenomenon, write the equation, find the function that solves it and draw the graph. flowchart LR phenomenon((FENÔMENO)) --> equation["EQUAÇÃODIFERENCIAL"] equation --> solution["FUNÇÃOSOLUÇÃO"] solution --> plottedGraph((GRÁFICO)) plottedGraph --> derivative{"DERIVADA"} derivative --> phenomenon derivative -. "localiza" .-> extrema["Pontos de máximoe de mínimo"] classDef phenomenonStyle fill:#e6f1ee,stroke:#397c75,color:#243448,stroke-width:2px classDef equationStyle fill:#eaf0f7,stroke:#58799b,color:#243448,stroke-width:2px classDef solutionStyle fill:#f8efd9,stroke:#b68b3c,color:#243448,stroke-width:2px classDef derivativeStyle fill:#f5e9e3,stroke:#9b6251,color:#243448,stroke-width:2px class phenomenon phenomenonStyle class equation,plottedGraph equationStyle class solution solutionStyle class derivative,extrema derivativeStyle The phenomenon. In 1678, Robert Hooke summarized the behavior of a spring in Latin: ut tensio, sic vis, “as extension, so strength.” A spring of constant $k$, stretched by $x$, pulls back with strength $F = -kx$.

September 29, 2026 Read